We have that c−1 (c(T )) = T and c(c−1 (U )) = U ; c(T ) is permutable with T , and commutes with any S ∈ L(H) which is permutable with T . 2 Definition. The operator c(T ) is the Cayley transform of T , and c−1 (U ) is the inverse Cayley transform of U . If E is a spectral projection of c(T ) corresponding to a Borel subset A of T, then E is permutable with T ; in fact, EU is a unitary on EH, and it is easily veriﬁed that c−1 (EU ), which is a self-adjoint operator on EH, agrees with T |EH . Furthermore, E commutes with any bounded operator which is permutable with T .